An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fraïssé limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.

We consider injective first-order interpretations that input and output trees of bounded height. The corresponding functions have polynomial output size, since a first-order interpretation can use a k -tuple of input nodes to represent a single output node. We prove that the equivalence problem for such functions is decidable, i.e. given two such interpretations, one can decide whether, for every input tree, the two output trees are isomorphic. We also give a calculus of typed functions and combinators which derives exactly injective first-order interpretations for unordered trees of bounded height. The calculus is based on a type system, where the type constructors are products, coproducts and a monad of multisets. Thanks to our results about tree-to-tree interpretations, the equivalence problem is decidable for this calculus. As an application, we show that the equivalence problem is decidable for first-order interpretations between classes of graphs that have bounded tree-depth. In all cases studied in this paper, first-order logic and mso have the same expressive power, and hence all results apply also to mso interpretations.

We introduce the countdown $μ$-calculus, an extension of the modal $μ$-calculus with ordinal approximations of fixpoint operators. In addition to properties definable in the classical calculus, it can express (un)boundedness properties such as the existence of arbitrarily long sequences of specific actions. The standard correspondence with parity games and automata extends to suitably defined countdown games and automata. However, unlike in the classical setting, the scalar fragment is provably weaker than the full vectorial calculus and corresponds to automata satisfying a simple syntactic condition. We establish some facts, in particular decidability of the model checking problem and strictness of the hierarchy induced by the maximal allowed nesting of our new operators.

Modal μ-calculus is a well-known formalism for describing properties of state-based transition systems. It can define properties such as "[in the current state] p holds, and there is a path where is holds again at some point in the future", where p comes from some fixed vocabulary of basic predicates. A formula of the classical μ-calculus refers only to finitely many basic predicates, which may sometimes seem restrictive. Real systems routinely operate on data coming from potentially infinite domains, such as numbers or character strings. Basic properties of such systems may reasonably include ones like "the number n was input", for every number n. It is then not clear how to say that "there exists a transition path where the currently input number is input again some time in the future" as a formula. Various modal formalisms have been proposed to model temporal properties of systems that refer to data coming from infinite domains. Here I focus on the modal μ-calculus with atoms, which is an extension of the classical calculus with features of nominal sets. There, basic predicates, formulas and models rely on atoms that come from some fixed infinite domain and can be tested for equality (or, in an extended variant, for some fixed order). I present a few variants of the modal μ-calculus with atoms, and describe their properties. As an example application, I show how to formulate the security property of the cryptographic Needham-Schroeder protocol, which relies on generating atomic nonces and comparing them for equality, and which famously fails due to a man-in-the-middle attack. Much of the material presented in this talk is drawn from [C. Eberhart and B. Klin, 2019; B. Klin and M. Łełyk, 2019; B. Klin and M. Łełyk, 2017].

Abstract We prove that if a data language and its complement are both recognized by nondeterministic register automata (without guessing), then they are also recognized by deterministic ones.

We develop a theory of vector spaces spanned by orbit-finite sets. Using this theory, we give a decision procedure for equivalence of weighted register automata, which are the common generalization of weighted automata and register automata for infinite alphabets. The algorithm runs in exponential time, and in polynomial time for a fixed number of registers. As a special case, we can decide, with the same complexity, language equivalence for unambiguous register automata, which improves previous results in three ways: (a) we allow for order comparisons on atoms, and not just equality; (b) the complexity is exponentially better; and (c) we allow automata with guessing.

$\omega$-clones are multi-sorted structures that naturally emerge as algebras for infinite trees, just as $\omega$-semigroups are convenient algebras for infinite words. In the algebraic theory of languages, one hopes that a language is regular if and only if it is recognized by an algebra that is finite in some simple sense. We show that, for infinite trees, the situation is not so simple: there exists an $\omega$-clone that is finite on every sort and finitely generated, but recognizes a non-regular language.

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters.

We study an extension of modal $\mu$-calculus to sets with atoms and we study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show expressive limitations of atom-enriched $\mu$-calculi, and explain how their expressive power depends on the structure of atoms used, and on the choice between basic or vectorial syntax.

We introduce an extension of modal mu-calculus to sets with atoms and study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show some limitations to the expressiveness of the calculus and argue that a naive way to remove these limitations results in a logic whose model checking is undecidable.

Labelled Markov processes are probabilistic versions of labelled transition systems. In general, the state space of a labelled Markov process may be a continuum. Logical characterizations of probabilistic bisimulation and simulation were given by Desharnais et al. These results hold for systems defined on analytic state spaces and assume that there are countably many labels in the case of bisimulation and finitely many labels in the case of simulation. In this paper, we first revisit these results by giving simpler and more streamlined proofs. In particular, our proof for simulation has the same structure as the one for bisimulation, relying on a new result of a topological nature. This departs from the known proof for this result, which uses domain theory techniques and falls out of a theory of approximation of Labelled Markov processes. Both our proofs assume the presence of countably many labels. We investigate the necessity of this assumption, and show that the logical characterization of bisimulation may fail when there are uncountably many labels. However, with a stronger assumption on the transition functions (continuity instead of just measurability), we can regain the logical characterization result, for arbitrarily many labels. These new results arose from a new game-theoretic way of understanding probabilistic simulation and bisimulation.

We present an Angluin-style algorithm to learn nominal automata, which are acceptors of languages over infinite (structured) alphabets. The abstract approach we take allows us to seamlessly extend known variations of the algorithm to this new setting. In particular we can learn a subclass of nominal non-deterministic automata. An implementation using a recently developed Haskell library for nominal computation is provided for preliminary experiments.

We investigate several variants of the homomorphism problem: given two relational structures, is there a homomorphism from one to the other? The input structures are possibly infinite, but definable by first-order interpretations in a fixed structure. Their signatures can be either finite or infinite but definable. The homomorphisms can be either arbitrary, or definable with parameters, or definable without parameters. For each of these variants, we determine its decidability status.

A format for well-behaved translations between structural operational specifications is derived from a notion of distributive law morphism, previously studied by Power and Watanabe.

First-order definable structures with atoms are infinite, but exhibit enough symmetry to be effectively manipulated. We study Constraint Satisfaction Problems (CSPs) where both the instance and the template are definable structures with atoms. As an initial step, we consider locally finite templates, which contain potentially infinitely many finite relations. We argue that such templates occur naturally in Descriptive Complexity Theory. We study CSPs over such templates for both finite and infinite, definable instances. In the latter case even decidability is not obvious, and to prove it we apply results from topological dynamics. For finite instances, we show that some central results from the classical algebraic theory of CSPs still hold: the complexity is determined by polymorphisms of the template, and the existence of certain polymorphisms, such as majority or Maltsev polymorphisms, guarantees the correctness of classical algorithms for solving finite CSP instances.

We study Turing machines over sets with atoms, also known as nominal sets. Our main result is that deterministic machines are weaker than nondeterministic ones; in particular, P≠NP in sets with atoms. Our main construction is closely related to the Cai-Furer-Immerman graphs used in descriptive complexity theory.

Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing lambda-terms modulo alpha-conversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming language, called Nlambda. The key idea is that it includes a native type for finite sets in the nominal sense. To illustrate the power of our language, we write short programs that process automata on data words.

This special issue of Mathematical Structures in Computer Science contains a selection of papers presented at three satellite events of CONCUR'09, which was held between 31 August and 5 September 2009 in Bologna (Italy). Specifically, it contains three papers from the 16th International Workshop on Expressiveness in Concurrency (EXPRESS'09), one paper from the 2nd Interaction and Concurrency Experience (ICE'09) and two papers from the 6th Workshop on Structural Operational Semantics (SOS'09).

Our motivating question is a My hill-Nerode theorem for infinite alphabets. We consider several kinds of those: alphabets whose letters can be compared only for equality, but also ones with more structure, such as a total order or a partial order. We develop a framework for studying such alphabets, where the key role is played by the automorphism group of the alphabet. This framework builds on the idea of nominal sets of Gabbay and Pitts, nominal sets are the special case of our framework where letters can be only compared for equality. We use the framework to uniformly generalize to infinite alphabets parts of automata theory, including decidability results. In the case of letters compared for equality, we obtain automata equivalent in expressive power to finite memory automata, as defined by Francez and Kaminski.

Final coalgebras capture system behaviours such as streams, infinite trees and processes. Algebraic operations on a final coalgebra can be defined by distributive laws (of a syntax functor Σ over a behaviour functor F). Such distributive laws correspond to abstract specification formats. One such format is a generalisation of the GSOS rules known from structural operational semantics of processes. We show that given an abstract GSOS specification ρ that defines operations σ on a final F-coalgebra, we can systematically construct a GSOS specification ρ that defines the pointwise extension σ of σ on a final FA-coalgebra. The construction relies on the addition of a family of auxiliary ‘buffer’ operations to the syntax. These buffer operations depend only on A, so the construction is uniform for all σ and F.

A novel, general approach is proposed to proving the compositionality of process equivalences on languages defined by structural operational semantics (SOS). The approach, based on modal logic, is inspired by the simple observation that if the set of formulas satisfied by a process can be derived from the corresponding sets for its subprocesses, then the logical equivalence is a congruence. Striving for generality, SOS rules are modeled categorically as bialgebraic distributive laws for some notions of process syntax and behaviour, and modal logics are modeled via coalgebraic polyadic modal logic. Compositionality is proved by providing a suitable notion of behaviour for the logic together with a dual distributive law, reflecting the one modeling the SOS specification. Concretely, the dual laws may appear as SOS-like rules where logical formulas play the role of processes, and their behaviour models logical decomposition over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats.